Γ-convergence and stochastic homogenisation of phase-transition functionals

Angewandte Mathematik, WWU Münster, Germany

^* Corresponding author: roberta.marziani@uni-muenster.de

Received: 27 June 2022
Accepted: 16 April 2023

Abstract

In this paper, we study the asymptotics of singularly perturbed phase-transition functionals of the form

ℱ_k(u) = 1/ε_k∫_Af_k(𝑥,u,ε_k∇u)d𝑥,

where u ∈ [0, 1] is a phase-field variable, ε_k > 0 a singular-perturbation parameter i.e., ε_k → 0, as k → +∞, and the integrands f_k are such that, for every x and every k, f_k(x, ·, 0) is a double well potential with zeros at 0 and 1. We prove that the functionals F_k Γ-converge (up to subsequences) to a surface functional of the form

ℱ_∞(u) = ∫_Su∩Af_∞(𝑥,𝜈_u)dH^n-1,

where u ∈ BV(A; {0, 1}) and f_∞ is characterised by the double limit of suitably scaled minimisation problems. Afterwards we extend our analysis to the setting of stochastic homogenisation and prove a Γ-convergence result for stationary random integrands.